In recent years, some investigations have been carried out to examine the assumptions like stationarity, symmetry and separability of spatio-temporal covariance function which would considerably simplify fitting a valid covariance model to the data by parametric and nonparametric methods. In this article, assuming a Gaussian random field, we consider the likelihood ratio separability test, a variety of nonparametric and spectral tests of symmetry and also separability of spatio-temporal covariance function. Comparing the tests of symmetry and separability in a level of scenarios, the best ones would be introduced.